Home » date » 2009 » Dec » 11 »

Exponential Smoothing

*The author of this computation has been verified*

R Software Module: /rwasp_exponentialsmoothing.wasp (opens new window with default values)

Title produced by software: Exponential Smoothing

Date of computation: Fri, 11 Dec 2009 09:49:18 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Dec/11/t1260550211zefiqg59ixvp6oy.htm/, Retrieved Fri, 11 Dec 2009 17:50:15 +0100

BibTeX entries for LaTeX users:

@Manual{KEY,
    author = {{YOUR NAME}},
    publisher = {Office for Research Development and Education},
    title = {Statistical Computations at FreeStatistics.org, URL http://www.freestatistics.org/blog/date/2009/Dec/11/t1260550211zefiqg59ixvp6oy.htm/},
    year = {2009},
}
@Manual{R,
    title = {R: A Language and Environment for Statistical Computing},
    author = {{R Development Core Team}},
    organization = {R Foundation for Statistical Computing},
    address = {Vienna, Austria},
    year = {2009},
    note = {{ISBN} 3-900051-07-0},
    url = {http://www.R-project.org},
}

Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

» Textbox « » Textfile « » CSV «

21790 13253 37702 30364 32609 30212 29965 28352 25814 22414 20506 28806 22228 13971 36845 35338 35022 34777 26887 23970 22780 17351 21382 24561 17409 11514 31514 27071 29462 26105 22397 23843 21705 18089 20764 25316 17704 15548 28029 29383 36438 32034 22679 24319 18004 17537 20366 22782 19169 13807 29743 25591 29096 26482 22405 27044 17970 18730 19684 19785 18479 10698 31956 29506 34506 27165 26736 23691 18157 17328 18205 20995 17382 9367 31124 26551 30651 25859 25100 25778 20418 18688 20424 24776 19814 12738 31566 30111 30019 31934 25826 26835 20205 17789 20520 22518 15572 11509 25447 24090 27786 26195 20516 22759 19028 16971 20036 22485 18730

Output produced by software:

Summary of computational transaction
Raw Input	view raw input (R code)
Raw Output	view raw output of R engine
Computing time	2 seconds
R Server	'Gwilym Jenkins' @ 72.249.127.135

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.230182393893731
beta	0
gamma	0.28444699539051

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	22228	21149.8295975033	1078.17040249674
14	13971	13588.7596295972	382.240370402844
15	36845	36445.3424185781	399.657581421903
16	35338	35450.8049382778	-112.804938277804
17	35022	35304.5430264437	-282.543026443716
18	34777	35144.8728998077	-367.872899807655
19	26887	29732.5753717209	-2845.57537172092
20	23970	27435.850287071	-3465.85028707102
21	22780	24233.9575488456	-1453.95754884561
22	17351	20598.4935099989	-3247.49350999893
23	21382	17937.4847989449	3444.51520105514
24	24561	26008.0581728100	-1447.05817281004
25	17409	19964.8424926678	-2555.84249266784
26	11514	12243.8474802573	-729.847480257307
27	31514	32055.8959657687	-541.895965768697
28	27071	30881.6624255383	-3810.66242553825
29	29462	29865.7841584855	-403.784158485501
30	26105	29671.5674365729	-3566.56743657292
31	22397	23992.4787650306	-1595.47876503061
32	23843	22126.0073240852	1716.99267591479
33	21705	20805.3919757416	899.60802425836
34	18089	17695.1369369274	393.863063072622
35	20764	17338.2804380436	3425.71956195636
36	25316	23895.2288282854	1420.77117171457
37	17704	18518.8460800047	-814.846080004656
38	15548	11769.6859606657	3778.31403933433
39	28029	33877.5975507525	-5848.59755075249
40	29383	30701.9994781931	-1318.99947819307
41	36438	31032.3782502449	5405.6217497551
42	32034	31392.211465989	641.788534010993
43	22679	26563.4959835361	-3884.49598353610
44	24319	24811.3793141501	-492.379314150086
45	18004	22654.0412966401	-4650.04129664013
46	17537	18095.0421221875	-558.042122187544
47	20366	18151.4172882810	2214.58271171902
48	22782	23925.1751122384	-1143.17511223840
49	19169	17686.5555439228	1482.44445607718
50	13807	12453.6112013796	1353.38879862038
51	29743	30846.6773732998	-1103.67737329980
52	25591	29769.2538182288	-4178.25381822879
53	29096	30777.7198476437	-1681.71984764368
54	26482	28636.2333948486	-2154.23339484863
55	22405	22810.3158587468	-405.315858746832
56	27044	22608.8620358893	4435.13796411073
57	17970	20741.2948231539	-2771.29482315392
58	18730	17567.7188372411	1162.28116275894
59	19684	18613.5536989637	1070.44630103632
60	19785	23317.8696718011	-3532.86967180106
61	18479	17306.6578088608	1172.34219113921
62	10698	12203.4019613282	-1505.40196132819
63	31956	27777.5295804939	4178.47041950614
64	29506	27291.5773317793	2214.42266822073
65	34506	30305.1720737296	4200.8279262704
66	27165	29329.6346858415	-2164.63468584151
67	26736	23677.1389526525	3058.86104734754
68	23691	25363.6832163915	-1672.68321639146
69	18157	20424.4120690273	-2267.41206902729
70	17328	18189.8409635505	-861.840963550465
71	18205	18742.3652604285	-537.365260428542
72	20995	21954.7512370020	-959.75123700203
73	17382	17561.6668823415	-179.666882341469
74	9367	11655.7105238571	-2288.71052385711
75	31124	27670.2016298888	3453.79837011123
76	26551	26652.8408876847	-101.840887684688
77	30651	29367.2392659885	1283.76073401151
78	25859	26578.4534003276	-719.45340032757
79	25100	22657.518093347	2442.48190665300
80	25778	23162.7872603926	2615.21273960745
81	20418	19228.2768087773	1189.7231912227
82	18688	18088.3098553626	599.6901446374
83	20424	19067.5760774308	1356.42392256924
84	24776	22776.4285230624	1999.57147693764
85	19814	18918.4361508756	895.56384912437
86	12738	12178.6161757851	559.38382421489
87	31566	32900.2593318825	-1334.25933188250
88	30111	29706.9363832271	404.063616772899
89	30019	33207.3254338703	-3188.32543387033
90	31934	28654.5017233966	3279.49827660344
91	25826	25967.3458825070	-141.345882507041
92	26835	25900.6731709862	934.32682901384
93	20205	20910.5442576052	-705.544257605234
94	17789	19128.1665817947	-1339.16658179469
95	20520	19847.8475332951	672.15246670493
96	22518	23590.1985439608	-1072.19854396078
97	15572	18844.7977293402	-3272.79772934022
98	11509	11516.4325405584	-7.43254055842954
99	25447	30202.2039985014	-4755.20399850137
100	24090	26846.3398832316	-2756.33988323155
101	27786	28492.0808521020	-706.080852102026
102	26195	26167.4788079740	27.5211920259644
103	20516	22528.9946074653	-2012.99460746532
104	22759	22231.5700946494	527.429905350604
105	19028	17623.021208704	1404.97879129598
106	16971	16406.9627957891	564.037204210872
107	20036	17844.0496883589	2191.95031164107
108	22485	21263.164670643	1221.83532935697
109	18730	16871.8810872562	1858.11891274379

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
110	11464.8978503417	9473.61492331017	13456.1807773733
111	28999.7388934272	25737.2725701882	32262.2052166662
112	27101.4466139435	23826.8113084822	30376.0819194048
113	29990.8291591272	26311.3368330068	33670.3214852476
114	27862.4977823095	24236.4293218185	31488.5662428005
115	23499.7975455901	20144.864914517	26854.7301766633
116	24286.4163462993	20696.8526206039	27875.9800719946
117	19377.3104256885	16164.6357435892	22589.9851077878
118	17547.887657666	14378.3324942246	20717.4428211074
119	19287.553675255	15739.3268582732	22835.7804922368
120	22052.6165491018	17974.4776577392	26130.7554404645
121	17459.4568282603	14524.2101107675	20394.7035457531

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/11/t1260550211zefiqg59ixvp6oy/1l2go1260550155.png (open in new window)

http://www.freestatistics.org/blog/date/2009/Dec/11/t1260550211zefiqg59ixvp6oy/1l2go1260550155.ps (open in new window)

http://www.freestatistics.org/blog/date/2009/Dec/11/t1260550211zefiqg59ixvp6oy/230at1260550155.png (open in new window)

http://www.freestatistics.org/blog/date/2009/Dec/11/t1260550211zefiqg59ixvp6oy/230at1260550155.ps (open in new window)

http://www.freestatistics.org/blog/date/2009/Dec/11/t1260550211zefiqg59ixvp6oy/3pba21260550155.png (open in new window)

http://www.freestatistics.org/blog/date/2009/Dec/11/t1260550211zefiqg59ixvp6oy/3pba21260550155.ps (open in new window)

Parameters (Session):

par1 = 12 ; par2 = periodic ; par3 = 0 ; par5 = 1 ; par7 = 1 ; par8 = FALSE ;

Parameters (R input):

par1 = 12 ; par2 = Triple ; par3 = multiplicative ; par5 = 1 ; par7 = 1 ; par8 = FALSE ;

R code (references can be found in the software module):

par1 <- as.numeric(par1)

if (par2 == 'Single') K <- 1

if (par2 == 'Double') K <- 2

if (par2 == 'Triple') K <- par1

nx <- length(x)

nxmK <- nx - K

x <- ts(x, frequency = par1)

if (par2 == 'Single') fit <- HoltWinters(x, gamma=0, beta=0)

if (par2 == 'Double') fit <- HoltWinters(x, gamma=0)

if (par2 == 'Triple') fit <- HoltWinters(x, seasonal=par3)

fit

myresid <- x - fit$fitted[,'xhat']

bitmap(file='test1.png')

op <- par(mfrow=c(2,1))

plot(fit,ylab='Observed (black) / Fitted (red)',main='Interpolation Fit of Exponential Smoothing')

plot(myresid,ylab='Residuals',main='Interpolation Prediction Errors')

par(op)

dev.off()

bitmap(file='test2.png')

p <- predict(fit, par1, prediction.interval=TRUE)

np <- length(p[,1])

plot(fit,p,ylab='Observed (black) / Fitted (red)',main='Extrapolation Fit of Exponential Smoothing')

dev.off()

bitmap(file='test3.png')

op <- par(mfrow = c(2,2))

acf(as.numeric(myresid),lag.max = nx/2,main='Residual ACF')

spectrum(myresid,main='Residals Periodogram')

cpgram(myresid,main='Residal Cumulative Periodogram')

qqnorm(myresid,main='Residual Normal QQ Plot')

qqline(myresid)

par(op)

dev.off()

load(file='createtable')

a<-table.start()

a<-table.row.start(a)

a<-table.element(a,'Estimated Parameters of Exponential Smoothing',2,TRUE)

a<-table.row.end(a)

a<-table.row.start(a)

a<-table.element(a,'Parameter',header=TRUE)

a<-table.element(a,'Value',header=TRUE)

a<-table.row.end(a)

a<-table.row.start(a)

a<-table.element(a,'alpha',header=TRUE)

a<-table.element(a,fit$alpha)

a<-table.row.end(a)

a<-table.row.start(a)

a<-table.element(a,'beta',header=TRUE)

a<-table.element(a,fit$beta)

a<-table.row.end(a)

a<-table.row.start(a)

a<-table.element(a,'gamma',header=TRUE)

a<-table.element(a,fit$gamma)

a<-table.row.end(a)

a<-table.end(a)

table.save(a,file='mytable.tab')

a<-table.start()

a<-table.row.start(a)

a<-table.element(a,'Interpolation Forecasts of Exponential Smoothing',4,TRUE)

a<-table.row.end(a)

a<-table.row.start(a)

a<-table.element(a,'t',header=TRUE)

a<-table.element(a,'Observed',header=TRUE)

a<-table.element(a,'Fitted',header=TRUE)

a<-table.element(a,'Residuals',header=TRUE)

a<-table.row.end(a)

for (i in 1:nxmK) {

a<-table.row.start(a)

a<-table.element(a,i+K,header=TRUE)

a<-table.element(a,x[i+K])

a<-table.element(a,fit$fitted[i,'xhat'])

a<-table.element(a,myresid[i])

a<-table.row.end(a)

}

a<-table.end(a)

table.save(a,file='mytable1.tab')

a<-table.start()

a<-table.row.start(a)

a<-table.element(a,'Extrapolation Forecasts of Exponential Smoothing',4,TRUE)

a<-table.row.end(a)

a<-table.row.start(a)

a<-table.element(a,'t',header=TRUE)

a<-table.element(a,'Forecast',header=TRUE)

a<-table.element(a,'95% Lower Bound',header=TRUE)

a<-table.element(a,'95% Upper Bound',header=TRUE)

a<-table.row.end(a)

for (i in 1:np) {

a<-table.row.start(a)

a<-table.element(a,nx+i,header=TRUE)

a<-table.element(a,p[i,'fit'])

a<-table.element(a,p[i,'lwr'])

a<-table.element(a,p[i,'upr'])

a<-table.row.end(a)

}

a<-table.end(a)

table.save(a,file='mytable2.tab')

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.

Software written by Ed van Stee & Patrick Wessa

Disclaimer

Information provided on this web site is provided "AS IS" without warranty of any kind, either express or implied, including, without limitation, warranties of merchantability, fitness for a particular purpose, and noninfringement. We use reasonable efforts to include accurate and timely information and periodically update the information, and software without notice. However, we make no warranties or representations as to the accuracy or completeness of such information (or software), and we assume no liability or responsibility for errors or omissions in the content of this web site, or any software bugs in online applications. Your use of this web site is AT YOUR OWN RISK. Under no circumstances and under no legal theory shall we be liable to you or any other person for any direct, indirect, special, incidental, exemplary, or consequential damages arising from your access to, or use of, this web site.

We may request personal information to be submitted to our servers in order to be able to:

personalize online software applications according to your needs
enforce strict security rules with respect to the data that you upload (e.g. statistical data)
manage user sessions of online applications
alert you about important changes or upgrades in resources or applications

We NEVER allow other companies to directly offer registered users information about their products and services. Banner references and hyperlinks of third parties NEVER contain any personal data of the visitor.

We do NOT sell, nor transmit by any means, personal information, nor statistical data series uploaded by you to third parties.

We carefully protect your data from loss, misuse, alteration, and destruction. However, at any time, and under any circumstance you are solely responsible for managing your passwords, and keeping them secret.

We store a unique ANONYMOUS USER ID in the form of a small 'Cookie' on your computer. This allows us to track your progress when using this website which is necessary to create state-dependent features. The cookie is used for NO OTHER PURPOSE. At any time you may opt to disallow cookies from this website - this will not affect other features of this website.

We examine cookies that are used by third-parties (banner and online ads) very closely: abuse from third-parties automatically results in termination of the advertising contract without refund. We have very good reason to believe that the cookies that are produced by third parties (banner ads) do NOT cause any privacy or security risk.

FreeStatistics.org is safe. There is no need to download any software to use the applications and services contained in this website. Hence, your system's security is not compromised by their use, and your personal data - other than data you submit in the account application form, and the user-agent information that is transmitted by your browser - is never transmitted to our servers.

As a general rule, we do not log on-line behavior of individuals (other than normal logging of webserver 'hits'). However, in cases of abuse, hacking, unauthorized access, Denial of Service attacks, illegal copying, hotlinking, non-compliance with international webstandards (such as robots.txt), or any other harmful behavior, our system engineers are empowered to log, track, identify, publish, and ban misbehaving individuals - even if this leads to ban entire blocks of IP addresses, or disclosing user's identity.

FreeStatistics.org is powered by